Properties

Label 2-91-1.1-c7-0-2
Degree $2$
Conductor $91$
Sign $1$
Analytic cond. $28.4270$
Root an. cond. $5.33170$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.95·2-s − 32.1·3-s − 79.6·4-s − 381.·5-s − 223.·6-s − 343·7-s − 1.44e3·8-s − 1.15e3·9-s − 2.65e3·10-s + 1.24e3·11-s + 2.56e3·12-s − 2.19e3·13-s − 2.38e3·14-s + 1.22e4·15-s + 147.·16-s − 1.24e4·17-s − 8.01e3·18-s + 7.19e3·19-s + 3.03e4·20-s + 1.10e4·21-s + 8.68e3·22-s + 1.43e4·23-s + 4.64e4·24-s + 6.74e4·25-s − 1.52e4·26-s + 1.07e5·27-s + 2.73e4·28-s + ⋯
L(s)  = 1  + 0.614·2-s − 0.687·3-s − 0.622·4-s − 1.36·5-s − 0.422·6-s − 0.377·7-s − 0.997·8-s − 0.527·9-s − 0.839·10-s + 0.282·11-s + 0.427·12-s − 0.277·13-s − 0.232·14-s + 0.938·15-s + 0.00902·16-s − 0.614·17-s − 0.324·18-s + 0.240·19-s + 0.849·20-s + 0.259·21-s + 0.173·22-s + 0.246·23-s + 0.685·24-s + 0.863·25-s − 0.170·26-s + 1.05·27-s + 0.235·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $1$
Analytic conductor: \(28.4270\)
Root analytic conductor: \(5.33170\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(0.4756972307\)
\(L(\frac12)\) \(\approx\) \(0.4756972307\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + 343T \)
13 \( 1 + 2.19e3T \)
good2 \( 1 - 6.95T + 128T^{2} \)
3 \( 1 + 32.1T + 2.18e3T^{2} \)
5 \( 1 + 381.T + 7.81e4T^{2} \)
11 \( 1 - 1.24e3T + 1.94e7T^{2} \)
17 \( 1 + 1.24e4T + 4.10e8T^{2} \)
19 \( 1 - 7.19e3T + 8.93e8T^{2} \)
23 \( 1 - 1.43e4T + 3.40e9T^{2} \)
29 \( 1 + 1.37e4T + 1.72e10T^{2} \)
31 \( 1 + 8.71e4T + 2.75e10T^{2} \)
37 \( 1 + 1.07e5T + 9.49e10T^{2} \)
41 \( 1 + 4.51e5T + 1.94e11T^{2} \)
43 \( 1 - 1.45e5T + 2.71e11T^{2} \)
47 \( 1 - 2.20e5T + 5.06e11T^{2} \)
53 \( 1 + 5.16e5T + 1.17e12T^{2} \)
59 \( 1 - 3.84e5T + 2.48e12T^{2} \)
61 \( 1 + 1.96e6T + 3.14e12T^{2} \)
67 \( 1 + 8.52e5T + 6.06e12T^{2} \)
71 \( 1 - 5.65e6T + 9.09e12T^{2} \)
73 \( 1 + 1.67e6T + 1.10e13T^{2} \)
79 \( 1 - 3.43e6T + 1.92e13T^{2} \)
83 \( 1 + 1.39e6T + 2.71e13T^{2} \)
89 \( 1 - 1.78e6T + 4.42e13T^{2} \)
97 \( 1 + 6.05e6T + 8.07e13T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.45911656093633057613989517445, −11.85406192107332739675112514884, −10.88563299585423016152835335666, −9.280537579206578424337624100187, −8.198719103162749338953868614580, −6.74590843995021626705715361940, −5.41728904599582029033098275295, −4.30617286344580801806645952794, −3.21270505148724694291320282933, −0.39768839727038919485728101541, 0.39768839727038919485728101541, 3.21270505148724694291320282933, 4.30617286344580801806645952794, 5.41728904599582029033098275295, 6.74590843995021626705715361940, 8.198719103162749338953868614580, 9.280537579206578424337624100187, 10.88563299585423016152835335666, 11.85406192107332739675112514884, 12.45911656093633057613989517445

Graph of the $Z$-function along the critical line